Non-Markovian diffusion equations and processes: analysis and simulations
aa r X i v : . [ m a t h - ph ] M a y Non-Markovian di(cid:27)usion equations and pro esses:analysis and simulationsAntonio MURA , Murad S. TAQQU and Fran es o MAINARDI . . Department of Mathemati s, Boston University, Boston, MA 02215, USAURL: http://math.bu.edu/people/murad/Revised Version: May 2008in press on Physi a A (2008), doi:10.1016/j.physa.2008.04.035Keywords: Non-Markovian pro esses, fra tional derivatives, anomalous di(cid:27)usion, subordination, fra tionalBrownian motion.Abstra t: In this paper we introdu e and analyze a lass of di(cid:27)usion type equations related to ertain non-Markovian sto hasti pro esses. We start from the forward drift equation whi h is made non-lo al in time by theintrodu tion of a suitable hosen memory kernel K ( t ) . The resulting non-Markovian equation an be interpretedin a natural way as the evolution equation of the marginal density fun tion of a random time pro ess l ( t ) . Wethen onsider the subordinated pro ess Y ( t ) = X ( l ( t )) where X ( t ) is a Markovian di(cid:27)usion. The orrespondingtime evolution of the marginal density fun tion of Y ( t ) is governed by a non-Markovian Fokker-Plan k equationwhi h involves the memory kernel K ( t ) . We develop several appli ations and derive the exa t solutions. We onsider di(cid:27)erent sto hasti models for the given equations providing path simulations.1 Introdu tionIn this introdu tion, we des ribe and motivate the themes developed in the paper. Histori al notes will bepresented in Se tion 2.Brownian motion B ( t ) , t ≥ , is a sto hasti pro ess with many properties. It is at the same time Gaussianand Markovian, has stationary in rements and is self-similar. A pro ess X ( t ) , t ≥ , is said to be self-similarwith self-similarity exponent H if, for all a ≥ , the pro esses X ( at ) , t ≥ , and a H X ( t ) , t ≥ , have the same(cid:28)nite-dimensional distributions. Brownian motion is self-similar with exponent H = 1 / . In ontrast, fra -tional Brownian motion B H ( t ) , t ≥ , is Gaussian, has stationary in rements, is self-similar with self-similarityexponent < H < , but is not Markovian, unless H = 1 / , in whi h ase the fra tional Brownian motionbe omes Brownian motion. When / < H < , the in rements of fra tional Brownian motion have long-rangedependen e [49℄.Be ause Brownian motion is Markovian with stationary in rements, its (cid:28)nite-dimensional distributions anbe obtained from the marginal density fun tion f B ( x, t ) = 1 √ πt e − x / t , x ∈ R (1)at time t ≥ . This density fun tion is the fundamental solution of the (cid:16)standard(cid:17) di(cid:27)usion equation: ∂ t u ( x, t ) = ∂ xx u ( x, t ) , (2)1hi h in integral form reads: u ( x, t ) = u ( x ) + Z t ∂ xx u ( x, s ) ds, u ( x ) = u ( x, . (3)Thus, f B ( x, t ) is a solution of Eq. (3) with u ( x ) = δ ( x ) , where δ ( x ) is the Dira delta distribution. We allow,throughout the paper, fun tions to be distributions.Remark 1.1. We follow the physi s onvention of not in luding the fa tor / in Eq. (2). Therefore, in thispaper, (cid:16)standard(cid:17) Brownian motion B ( t ) , t ≥ , is su h that, for ea h time t ≥ , B ( t ) ∼ N (0 , t ) . The (cid:16)tilde(cid:17)notation X ∼ f X ( x ) indi ates that the random variable X has the probability density fun tion f X ( x ) .Our goal is to extend Eq. (3) to non-Markovian settings. We will onsider non-lo al, fra tional and stret hedmodi(cid:28) ations of the di(cid:27)usion equation. These modi(cid:28)ed equations will be alled Non-Markovian di(cid:27)usion equa-tions, be ause, while they originate from a di(cid:27)usion equation, the orresponding pro ess, whose probabilitydensity fun tion is a solution of these modi(cid:28)ed equations, will be typi ally non-Markovian.To motivate the modi(cid:28) ations, onsider (cid:28)rst the non-random pro ess l ( t ) = t , t ≥ , whi h depi ts anon-random linear time evolution and let f l ( τ, t ) denote its density fun tion at time t . Therefore one has f l ( τ, t ) = δ ( τ − t ) where δ ( x ) is the Dira distribution. It is natural to interpret f l ( τ, t ) as the fundamentalsolution of the standard forward drift equation: ∂ t u ( τ, t ) = − ∂ τ u ( τ, t ) , τ, t ≥ , (4)whi h in integral form reads: u ( τ, t ) = u ( τ ) − Z t ∂ τ u ( τ, s ) ds, u ( τ ) = u ( τ, . (5)The general solutions are of the form u ( τ, t ) = u ( τ − t ) and thus, when u ( τ ) = δ ( τ ) , the solution of Eq. (4)is indeed u ( τ, t ) = δ ( τ − t ) . Observe that the variable τ ≥ plays the role of a spa e variable.We will onsider the following generalization of the forward drift equation (5) u ( τ, t ) = u ( τ ) − Z t K ( t − s ) ∂ τ u ( τ, s ) ds, τ, t ≥ , (6)where K ( t ) , with t ≥ , is a suitable kernel hosen su h that the fundamental solution of Eq. (6) is a probabilitydensity fun tion at ea h t ≥ . We refer to Eq. (6) as the non-Markovian forward drift equation.The presen e of the memory kernel K in Eq. (6) suggests a orresponding modi(cid:28) ation of the di(cid:27)usionequation (3). Namely, we will onsider the equation: u ( x, t ) = u ( x ) + Z t K ( t − s ) ∂ xx u ( x, s ) ds, x ∈ R , t ≥ . (7)Its fundamental solution turns out to be: f ( x, t ) = Z ∞ G ( x, τ ) h ( τ, t ) dτ, (8)where G ( x, t ) = 1 √ πt exp( − x / t ) , (9)and h ( τ, t ) is the fundamental solution of Eq. (6). 2he solution (8) is a marginal (one-point) probability density fun tion. We will onsider di(cid:27)erent randompro esses whose marginal probability density fun tion oin ides with it. As illustration, onsider the followingexamples1.Example 1.1. If we hoose: K ( t ) = t − / √ π , t ≥ , (10)then we have, see Eq. (65 and Eq. (70): h ( τ, t ) = 1 √ πt exp (cid:18) − τ t (cid:19) , τ ≥ , t ≥ , (11)as the fundamental solution of Eq. (6). Now onsider the pro ess D ( t ) = B ( l ( t )) , t ≥ , (12)where B is a (cid:16)standard(cid:17) Brownian motion and l ( t ) ≥ is a random time- hange (not ne essarily in reasing),independent of B , whose marginal density fun tion is given by h ( τ, t ) . One possible hoi e for the random timepro ess is simply: l ( t ) = | b ( t ) | , t ≥ , where b ( t ) , t ≥ , is a (cid:16)standard(cid:17) Brownian motion [9,18℄. Su h a random time pro ess l ( t ) , t ≥ , is self-similarof order H = 1 / .2 Let now B ( t ) , t ≥ , be another (cid:16)standard(cid:17) Brownian motion independent of b ( t ) . Thus,the pro ess (see also [3℄) D ( t ) = B ( | b ( t ) | ) , t ≥ , (13)has marginal density de(cid:28)ned by Eq. (8) with h ( τ, t ) given by Eq. (11).But, D ( t ) is not the only pro ess with density fun tion f ( x, t ) , given by Eq. (8). For example, the pro ess Y ( t ) = p | b (1) | B / ( t ) , t ≥ , (14)where B / is an independent fra tional Brownian motion with self-similarity exponent H = 1 / , has the sameone-dimensional probability density fun tions as the previous pro ess D ( t ) , t ≥ , see Eq. (40) with β = 1 / .Example 1.2. The fra tional Brownian motion in Eq. (14) has a self-similarity exponent H < / . Thein rements of su h a pro ess are known to be negatively orrelated [31, 32, 49℄. To allow for the presen e offra tional Brownian motion B H ( t ) with < H < , we introdu e a se ond (non-random) time- hange t → g ( t ) ,where g (0) = 0 and g ( t ) is smooth and in reasing, that is we onsider the non-Markovian di(cid:27)usion equation u ( x, t ) = u ( x ) + Z t g ′ ( s ) K ( g ( t ) − g ( s )) ∂ xx u ( x, s ) ds. (15)whose fundamental solution is now: f ( x, t ) = Z ∞ G ( x, τ ) h ( τ, g ( t )) dτ, (16)where h is the fundamental solution of Eq. (6). If K ( t ) is as in Eq. (10) and g ( t ) = t α , with < α < , thenthe pro esses: D ( t ) = B ( | b ( t α ) | ) , t ≥ ,Y ( t ) = p | b (1) | B α/ ( t ) , t ≥ , have a marginal density fun tion de(cid:28)ned by Eq. (16) with h ( τ, t ) as in Eq. (11), whi h is the fundamentalsolution of Eq. (15). In this ase Y ( t ) is de(cid:28)ned through an independent fra tional Brownian motion B α/ withHurst's parameter H = α/ and thus < H < . This is a spe ial ase of Eq. (78).1In these examples we refer to fa ts whi h are justi(cid:28)ed later in the paper through forward referen es. The reader may want tofo us at this point only on the examples and ignore the referen es.2 Another possible hoi e for a random time pro ess with marginal density given by Eq. (11) is the lo al time in zero of a(cid:16)standard(cid:17) Brownian motion [4℄. In this ase the time- hange pro ess l ( t ) is in reasing.3he pre eding examples illustrate the themes pursued in the paper. We will fo us, however, not only onpower-like kernels su h as those de(cid:28)ned in Eq. (10), but also on exponential-like kernels su h as: K ( t ) = e − at , a ≥ . (17)We also onsider what happens when the Brownian motion B ( t ) , t ≥ , is repla ed by a more general linear(time-homogeneous) di(cid:27)usion Q ( t ) , t ≥ , governed by the Fokker-Plan k equation3, ∂ t u ( x, t ) = P x u ( x, t ) , (18)where P x is a linear operator independent of t a ting on the variable x ∈ R . In other words we onsider thenon-Markovian di(cid:27)usion equation: u ( x, t ) = u ( x ) + Z t g ′ ( s ) K ( g ( t ) − g ( s )) P x u ( x, s ) ds. (19)We show that its fundamental solution is: f ( x, t ) = Z ∞ G ( x, τ ) h ( τ, g ( t )) dτ, (20)where G ( x, t ) is the fundamental solution of Eq. (19), while h ( τ, t ) is the fundamental solution of Eq. (6). Wealso provide expli it solutions when P x is the di(cid:27)erential operator asso iated with Brownian motion with drift,when it is asso iated with Geometri Brownian motion and when the kernel K ( t ) is the power kernel and theexponential kernel.In order not to dwell on te hni alities, we suppose impli itly, throughout the paper, that we have su(cid:30) ientregularity onditions, to justify the algebrai manipulations that are performed. The paper is organized asfollows:ˆ Histori al notes are presented in Se tion 2.ˆ In Se tion 3 we study the non-Markovian forward drift equation (6) and its orresponding random timepro ess l ( t ) . We derive suitability onditions on the kernel K ( t ) . We end the se tion by noting that aself-similar time- hange pro ess, for instan e with self-similarity parameter H = β , requires the hoi e K ( t ) = Ct β − / Γ( β ) with < β ≤ .ˆ In Se tion 4 we study the non-Markovian di(cid:27)usion equation (15) and its solutions, and we dis uss itsvarious sto hasti interpretations.ˆ In Se tion 5 we illustrate the fa t that the sto hasti representation is not unique.ˆ In Se tion 6 we study the more general non-Markovian Fokker-Plan k equation and derive its solution Eq.(20).ˆ In Se tion 7 we go thorough several examples with P x u ( x, t ) = ∂ xx u ( x, t ) , that is, when the underlyingdi(cid:27)usion pro es is Brownian motion. We onsider non-Markovian di(cid:27)usion equations, asso iated with the β -power kernel K ( t ) = t β − / Γ( β ) , < β ≤ , and with the exponential-de ay kernel K ( t ) = e − at , a ≥ .We also onsider di(cid:27)erent hoi es of the deterministi s aling fun tion g ( t ) , for example a logarithmi times ale g ( t ) = log( t + 1) is onsidered.ˆ In Se tion 8 we fo us on appli ations when the underlying di(cid:27)usion pro ess is not standard Brownianmotion. We onsider the ase of Brownian motion with drift and Geometri Brownian motion and westudy the orresponding equations with the β -power kernel and the exponential-de ay kernel.ˆ Se tion 9 ontains a summary and on luding remarks.3Also known as the forward Kolmogorov equation. 4 Histori al notesNon-Markovian equations like Eq. (7), or more generally Eq. (19), are often en ountered when studying physi alphenomena related to relaxation and di(cid:27)usion problems in omplex systems (see Srokowsky [47℄ for examples).Equations of the type (7) have been studied for example by Kolsrud [22℄. He obtained Eq. 8), but withoutproviding spe i(cid:28) examples. A similar study was done by Wyss [51℄ who, however, fo used only on power-likekernels K ( t ) = Ct β − .Sokolov [45℄ (see also Srokowsky [47℄), studied the non-Markovian equation ∂ t P ( x, t ) = Z t k ( t − s ) L x P ( x, s ) ds, (21)where L x is a linear operator a ting on the variable x . He provided a formal solution in the form of Eq. 20).Observe, however, that our equation (19) di(cid:27)ers from Eq. (21), not only by the presen e of the s aling fun tion g ( t ) , but also by the hoi e of the memory kernel. Our kernel K ( t ) and Sokolov's kernel k ( t ) are related by theequation: K ( t ) = Z t k ( s ) ds ⇒ e K ( s ) = e k ( s ) /s, s > , (22)where the tilde indi ates the Lapla e transform, see Eq. (25). The suitability onditions for these memorykernels are thus not the same (these onditions are developed in Se tion 3). For example, onsider the simpleexponential-de ay kernel e − at , a ≥ . This hoi e of the kernel is (cid:16)safe(cid:17) in the ontext of Eq. (19), i.e. for the hoi e K ( t ) = e − at , but is (cid:16)dangerous(cid:17) if one onsiders Eq. (21) with the kernel k ( t ) = e − at . In the ase ofEq. (19), the exponential-de ay kernel orresponds to a system for whi h non-lo al memory e(cid:27)e ts are initiallynegligible. In fa t, K ( t ) = e − at → as t → and thus the system appears Markovian at small times. On theother hand, the hoi e k ( t ) = e − at orresponds to the kernel K ( t ) = a − (1 − e − at ) whi h for small times behaveslike t . In this ase Sokolov [45℄ noti ed that the orresponding equations are only reasonable in a restri teddomain of the model parameters and for ertain initial and boundary onditions.Our starting point is di(cid:27)erent from that of the previous authors. Instead of starting dire tly from the Fokker-Plan k equations (18), we start from the forward drift equation (5) whi h is then generalized by introdu ing amemory kernel K ( t ) , Eq. (6). One is then naturally led to the non-Markovian di(cid:27)usion equations (15 and (19)after the introdu tion of the s aling fun tion g ( t ) . In fa t, in spe i(cid:28) ases, it is sometimes simpler to solve (cid:28)rstthe non-Markovian forward drift equation (6) and then use the solution to solve the non-Markovian di(cid:27)usionequation (15) or (19) by using (16) or (20). The form of the solution (16) or (20) has now a ready-made interpre-tation. For example, in Eq. (20) the fun tion G ( x, t ) is the fundamental solution of the Markovian equation (18)and the fun tion h ( τ, t ) is the fundamental solution of the non-Markovian equation (6) and it is these two solu-tions that ontribute to Eq. (20) whi h is the fundamental solution of the non-Markovian di(cid:27)usion equation (19).Furthermore, the form (16) or (20) has a natural interpretation in terms of subordinated pro esses, see Eq.(12). A ording to Whitmore and Lee [23℄, the term (cid:16)subordination(cid:17) was introdu ed by Bo hner [5, 6℄. It refersto pro esses of the form Y ( t ) = X ( l ( t )) , t ≥ , where X ( t ) , t ≥ , is a Markov pro ess and l ( t ) , t ≥ , is a(non-negative) random time pro ess independent of X . The marginal distribution of the subordinated pro essis learly: f Y ( x, t ) = Z ∞ f X ( x, τ ) f l ( τ, t ) dτ, t ≥ , x ∈ R , (23)where f X ( x, t ) and f l ( τ, t ) represent the marginal density fun tions of the pro esses X and l . Therefore, Eq.(16) or Eq. (20) an be interpreted in terms of subordinated pro esses, with Eq. (6) hara terizing the randomtime pro ess l ( t ) and Eq. (18) hara terizing the Markov parent pro ess X ( t ) .5he sto hasti interpretation through subordinated pro esses, (cid:28)rst suggested by Kolsrud, is very naturalbe ause Y ( t ) = X ( l ( t )) has a dire t physi al interpretation. For example, in equipment usage, X ( t ) an be thestate of a ma hine at time t and l ( t ) the e(cid:27)e tive usage up to time t . In an e onometri study, X ( t ) may bea model for the pri e of a sto k at time t . If l ( t ) measures the total e onomi a tivity up to time t , the pri eof the sto k at time t should not be des ribed by X ( t ) but by the subordinated pro ess Y ( t ) = X ( l ( t )) . Theresulting subordinated pro ess Y ( t ) is in general non-Markovian. In this way, the non-lo al memory e(cid:27)e ts areattributable to the random time pro ess l ( t ) and to its dynami s whi h is in general non-lo al in time, see Eq. (6).Note, however, that the solution of Eq. (19) represents only the marginal (one-point) density fun tion of thepro ess and therefore annot hara terize the full sto hasti stru ture of the pro ess. As we note in the paper,there are also pro esses that are not subordinated pro esses that serve as sto hasti models for non-Markoviandi(cid:27)usion equations like Eq. (19) or Eq. (21).For example, onsider in Eq. (7) the β -power kernel K ( t ) = t β − / Γ( β ) , with < β ≤ . From a sto hasti point of view, the fundamental solution of this equation, also alled the time-fra tional di(cid:27)usion equationof order β , an be interpreted as the marginal density fun tion of a self-similar sto hasti pro esses withparameter H = β/ . This pro ess, for example, an be taken to be a subordinated pro ess Y ( t ) = B ( l ( t )) ,with a suitable hoi e of the random time l . In Kolsrud [22℄, the random time l is taken to be related tothe lo al time of a d = 2(1 − β ) -dimensional fra tional Bessel pro ess, while in Meers haert et al. [34℄ (seealso [1, 15(cid:21)17, 21, 40, 48℄), in the ontext of a Continuous Time Random Walk (CTRW), it is hosen to be theinverse of the totally skewed stri tly β -stable pro ess. The interested reader is referred to the wide literature on erning the relationship between CTRW and non-Markovian di(cid:27)usion equations and its appli ations. Seefor instan e, [2, 13, 14, 19, 28, 35, 36, 41, 42, 50, 52℄ and referen es therein.S hneider [43℄, moreover, in a very general mathemati al onstru tion, introdu ed the so- alled Grey Brow-nian motion. This pro ess is a self-similar pro ess with stationary in rements whi h, as turns out, an berepresented by Y ( t ) = Λ β B H ( t ) , t ≥ , where B H is a fra tional Brownian motion with H = β/ and Λ β is asuitable hosen random variable independent of B H (see Mura et al. for details [38, 39℄). This pro ess has amarginal density fun tion that evolves in time a ording to the time-fra tional di(cid:27)usion equation of order β . Inthis ase the non-Markovian property is due to the presen e of the fra tional Brownian motion. As we show inthe paper, long-range dependen e an be made to appear through the time-s aling fun tion g ( t ) , see Eq. (15)and Example 1.2. Figures 4, 5 and 6 display traje tories of the pro esses D ( t ) and Y ( t ) and orrespondingdensity fun tions.3 The non-Markovian forward drift equationWe start with the following generalization of Eq. (5), namely: u ( τ, t ) = u ( τ ) − Z t K ( t − s ) ∂ τ u ( τ, s ) ds, τ, t ≥ , (24)where K ( t ) , with t ≥ , is a suitable hosen kernel. We then hoose a random time pro ess l ( t ) su h that,for ea h t ≥ , its marginal density f l ( τ, t ) is the fundamental solution of Eq. (24). Observe that Eq. (24) is(cid:16)non-lo al(cid:17) be ause u ( τ, t ) involves u ( τ, s ) at all ≤ s ≤ t . Equation (24) will be alled non-Markovian forwarddrift equation, see Se tion 1, Eq. (6).It is onvenient to work with Lapla e transforms. We indi ate by L { ϕ ( x, t ); t, s } the Lapla e transform ofthe fun tion ϕ with respe t to t evaluated in s ≥ , namely: L { ϕ ( x, t ); t, s } = Z ∞ e − ts ϕ ( x, t ) dt, s ≥ . (25)If the fun tion ϕ depends only on the variable t we write simply e ϕ ( s ) , be ause in this ase there is no ambiguity on erning the integration variable. In parti ular we let e K ( s ) denote the Lapla e transform of the kernel K .6roposition 3.1. Let f l ( τ, t ) denote the fundamental solution of Eq. (24). Then, L { f l ( τ, t ); t, s } = 1 s e K ( s ) exp − τ e K ( s ) ! , τ, s ≥ , (26)and zero for τ < .Proof: we take the Lapla e transform with respe t to the variable t in Eq. (24): ∂ τ e u ( τ, s ) = u ( τ ) s e K ( s ) − e u ( τ, s ) e K ( s ) , (27)thus Eq. (26) is a solution, in the distributional sense, when u ( τ ) = δ ( τ ) . Indeed the general solution ofEq.27) with u ( τ ) = δ ( τ ) is: ϕ ( τ, s ) = θ ( τ ) s e K ( s ) exp − τ e K ( s ) ! + C exp − τ e K ( s ) ! , τ ∈ R , where C is a real onstant and where θ ( x ) = (cid:26) , x ≥ , , x < (28)is the Heaviside's step fun tion. Sin e we require ϕ ( τ, t ) = 0 for τ < , we get C = 0 i.e. Eq. (26). (cid:3) K We must hoose the kernel K su h that the fundamental solution of Eq. (24) is a probability density in τ ≥ .We observe that if f l ( τ, t ) satis(cid:28)es Eq. (24) and Eq. (26), then it is automati ally normalized for ea h t ≥ .In fa t, for a fun tion ϕ ( x, t ) for whi h it is always possible to hange the order of integration, one has: Z R ϕ ( x, t ) dx = 1 ⇐⇒ Z R e ϕ ( x, s ) dx = s − . (29)Sin e Eq. (26) satis(cid:28)es the right-hand side of Eq. (29), we get R R + f l ( τ, t ) dτ = 1 . One still needs, however, to hoose the kernel K su h that f l ( τ, t ) ≥ for all τ, t ≥ .In order to get a suitable ondition on the kernel K , we make use of the notion of ompletely monotonefun tion. Re all that a fun tion ϕ ( t ) is ompletely monotone if it is non-negative and possesses derivatives ofany order and: ( − k d k dt k ϕ ( t ) ≥ , t > , k ∈ Z + = { , , , . . . } . (30)We observe that as t → , the limit of d k ϕ ( t ) /dt k may be (cid:28)nite or in(cid:28)nite. Typi al non-trivial examples are ϕ ( t ) = exp( − at ) , with a > , ψ ( t ) = 1 /t and φ ( t ) = 1 / (1 + t ) . It is easy to show that if ϕ and ψ are ompletelymonotone then their produ t ϕψ is as well. Moreover, if ϕ is ompletely monotone and ψ is positive with (cid:28)rstderivative ompletely monotone then the fun tion ϕ ( ψ ) is ompletely monotone.We have the following hara terization of ompletely monotone fun tions [10℄:Lemma 3.1. A fun tion ϕ ( s ) , de(cid:28)ned on the positive real line, is ompletely monotone if and only if is of theform: ϕ ( s ) = Z ∞ e − ts F ( dt ) , s ≥ , where F is a (cid:28)nite or in(cid:28)nite non-negative measure on the positive real semi-axis.Hen e, to ensure that f l ( τ, t ) ≥ for all τ, t ≥ , it is enough to require that the fun tion de(cid:28)ned in Eq.(26) must be ompletely monotone, as a fun tion of s , for any τ ≥ , and thus that the kernel K satis(cid:28)es thefollowing: 7uitability onditions1. s e K ( s ) is positive with (cid:28)rst derivative ompletely monotone,2. / e K ( s ) is positive with (cid:28)rst derivative ompletely monotone.Indeed, we an view Eq. 26) as the produ t of the two ompletely monotone fun tions /u and exp( − τ u ) , the(cid:28)rst evaluated at u = s e K ( s ) and the se ond evaluated at u = 1 / e K ( s ) .3.2 ExamplesExample 3.1 ( β -power kernel). If we hoose: K ( t ) = t β − Γ( β ) , we get e K ( s ) = s − β . In this ase s e K ( s ) = s − β is positive and has (cid:28)rst derivative (1 − β ) s − β ompletelymonotone if and only if < β ≤ . Moreover, / e K ( s ) = s β is positive with (cid:28)rst derivative βs β − ompletelymonotone if and only if < β ≤ . Therefore, a good hoi e for the kernel K is: K ( t ) = t β − Γ( β ) , < β ≤ . (31)Example 3.2 (Exponential-de ay kernel). Choosing: K ( t ) = exp( − at ) , a ≥ , (32)we get s e K ( s ) = s/ ( s + a ) whi h is positive with (cid:28)rst derivative a ( s + a ) − ompletely monotone for any a ≥ .Moreover, / e K ( s ) = ( s + a ) is positive if a ≥ with (cid:28)rst derivative ompletely monotone.Example 3.3 ( β -power with exponential-de ay kernel). Choosing: K ( t ) = t β − Γ( β ) exp( − at ) , < β ≤ , a ≥ , (33)we have e K ( s ) = ( s + a ) − β . Therefore, s e K ( s ) = s ( s + a ) − β whi h is positive if a ≥ with (cid:28)rst derivative ( s + a ) − β (1 − βs ( s + a ) − ) ompletely monotone if < β ≤ . Moreover, / e K ( s ) = ( s + a ) β is positive if a ≥ with (cid:28)rst derivative β ( s + a ) β − ompletely monotone if < β ≤ .The following theorem states that a self-similar random time pro ess l ( t ) , t ≥ , is asso iated with the kernel K ( t ) in Example 3.1:Theorem 3.1. If the time- hange pro ess l ( t ) , t ≥ , is self-similar (for instan e of order H = β ), withmarginal probability density f l ( τ, t ) satisfying Eq. (26), then we must have: K ( t ) = C t β − Γ( β ) , < β ≤ , (34)for some positive onstant C .Proof: The self-similarity ondition entails that for any τ, t ≥ and for any a > : a − β f l ( a − β τ, t ) = f l ( τ, at ) . If we take the Lapla e transform and set e f ( τ, s ) = L{ f l ( τ, t ); t, s } , we have: a − β e f l ( a − β τ, s ) = 1 a e f l (cid:16) τ, sa (cid:17) . τ, s ≥ and a > : a − β e K ( s ) exp − a − β τ e K ( s ) ! = 1 e K ( sa ) exp − τ e K ( sa ) ! . Sin e this relation is valid for any hoi e of τ ≥ and s ≥ , putting τ = 0 and s = a , we get: a − β e K ( a ) = 1 e K (1) . Thus, for any a > : e K ( a ) = e K (1) a − β , whi h is the Lapla e transform ofEq. 34). If we add moreover the ondition of omplete monotoni ity we (cid:28)nd: < β ≤ as indi ated in Example 3.1. (cid:3) h ( τ, t ) and de(cid:28)ned by Eq. (26).2. The fundamental solution G ( x, t ) , de(cid:28)ned by Eq. (9), of the standard di(cid:27)usion equation whi h is theone-dimensional density of the (cid:16)standard(cid:17) Brownian motion.The following theorem ombines these two ingredients and provides the fundamental solution of a orrespondingnon-Markovian di(cid:27)usion equation.Theorem 4.1. Let h ( τ, t ) denote the fundamental solution of Eq. 24), so that by Proposition 3.1, one has: L { h ( τ, t ); t, s } = 1 s e K ( s ) exp − τ e K ( s ) ! , τ, s ≥ , (35)for a suitable hoi e of K . Let g be a stri tly in reasing fun tion with g (0) = 0 and let G ( x, t ) be de(cid:28)ned by Eq.(9). Then, f ( x, t ) = Z ∞ G ( x, τ ) h ( τ, g ( t )) dτ, (36)is the fundamental solution of the non-Markovian di(cid:27)usion equation: u ( x, t ) = u ( t ) + Z t g ′ ( s ) K ( g ( t ) − g ( s )) ∂ xx u ( x, s ) ds. (37)Proof: see Se tion 6. (cid:3) We have immediately the following:Corollary 4.1. If H ( x, t ) is a solution of the standard di(cid:27)usion equation with initial ondition H ( x,
0) = u ( x ) ,then the fun tion: u ( x, t ) = Z ∞ H ( x, τ ) h ( τ, g ( t )) dτ (38)is a solution of Eq. (37). 9roof: If, for any t ≥ , the fun tion f ( x, t ) de(cid:28)ned in Eq. (36) is the fundamental solution of Eq. (37) then ageneral solution is given by: u ( x, t ) = Z R f ( x − y, t ) u ( y ) dy = Z R Z ∞ G ( x − y, τ ) u ( y ) h ( τ, g ( t )) dτ dy = Z ∞ (cid:18)Z R G ( x − y, τ ) u ( y ) dy (cid:19) h ( τ, g ( t )) dτ = Z ∞ H ( x, τ ) h ( τ, g ( t )) dτ. (cid:3) We observe that:1. The equation (35) states that h ( τ, t ) is the fundamental solution of Eq. (24).2. While G ( x, t ) is the fundamental solution of the standard di(cid:27)usion equation obtained when u ( x ) = δ ( x ) ,the general solution, denoted H ( x, t ) in the above theorem, results from a general initial ondition u ( x ) .Many physi al phenomena, espe ially related to relaxation pro esses in omplex systems, are des ribed by non-Markovian (cid:16)master equations(cid:17) like Eq. (37). K ( t ) is a memory kernel and g ( t ) is just a (cid:16)time-s aling(cid:16) fun tion.Su h equations are often argued by phenomenologi al onsiderations and an be more or less rigorously derivedstarting from a mi ros opi des ription [7, 20, 47, 53℄.5 The sto hasti representation is not uniqueThe solution of the non-Markovian di(cid:27)usion equation an be viewed as the marginal density fun tion of thesubordinated pro ess, see Eq. (12) D ( t ) = B ( l ( g ( t ))) , t ≥ , sin e its marginal density is: f D ( x, t ) = Z ∞ G ( x, τ ) f l ( τ, g ( t )) dτ. Here, for ea h t ≥ , D ( t ) ∼ f D ( x, t ) , B ( t ) ∼ G ( x, t ) and l ( t ) ∼ f l ( τ, t ) . In the notation of Theorem 4.1, wehave f D ( x, t ) = f ( x, t ) and f l ( τ, t ) = h ( τ, t ) . The Lapla e transform of f l ( τ, t ) with respe t to t is given by Eq.(35).This sto hasti representation is not unique (see Example 1.1, Example 1.2 and examples below). Indeed,the non-Markovian di(cid:27)usion equation hara terizes only the marginal, that is one-point, probability densityfun tion. However, pro esses with a di(cid:27)erent dependen e stru ture an have the same marginal density f ( x, t ) .Additional requirements ould be imposed so as to spe ify the sto hasti model more pre isely.Example 5.1. If we require the random time pro ess l β ( t ) , t ≥ , to be self-similar of order β , then in view ofTheorem 3.1, the kernel must be hosen as in Eq. (34) and we must have < β ≤ . We will study this asemore in details in Se tion 7. Here we just observe that if we onsider a (cid:16)standard(cid:17) fra tional Brownian motion B β/ of order β/ , then f ( x, t ) is also the marginal distribution of Y ( t ) = q l β (1) B β/ ( t ) , (39)where B β/ ( t ) is assumed to be independent of l β (1) .In fa t, be ause l β ( t ) , t ≥ , is self-similar of order H = β , one has: D ( t ) = B ( l β ( t )) = d q l β ( t ) B (1) = d q l β (1) t β/ B (1) = d q l β (1) t β/ B β/ (1) = d q l β (1) B β/ ( t ) = Y ( t ) , (40)where = d denotes here the equality of the marginal distributions.Both D ( t ) , t ≥ , and Y ( t ) , t ≥ , are self-similar pro esses with Hurst's exponent H = β/ . However, while Y ( t ) , t ≥ , has always stationary in rements, this is not in general true in the ase of the pro ess D ( t ) , t ≥ .10 Non-Markovian Fokker-Plan k equationWe onsidered up until now pro esses of the type B ( l ( g ( t ))) , where B is a (cid:16)standard(cid:17) Brownian motion. Whathappens if we repla e B by a more general di(cid:27)usion? Namely, what happens if instead of starting with thestandard di(cid:27)usion equation (2) we start with a more general Markovian Fokker-Plan k equation: ∂ t u ( x, t ) = P x u ( x, t ) , x ∈ R , t ≥ , (41)where P x is a linear operator, independent of t , a ting on the variable x ? We have the following generalizationof Theorem 4.1:Theorem 6.1. Suppose that h ( τ, t ) is a probability density fun tion satisfyingEq. 26) L { h ( τ, t ); t, s } = 1 s e K ( s ) exp − τ e K ( s ) ! , τ, s ≥ , (42)for a suitable hoi e of K . Let g be a stri tly in reasing fun tion with g (0) = 0 and G ( x, t ) be the fundamentalsolution of Eq. (41). Then the fundamental solution of the integral equation: u ( x, t ) = u ( t ) + Z t g ′ ( s ) K ( g ( t ) − g ( s )) P x u ( x, s ) ds (43)is f ( x, t ) = Z ∞ G ( x, τ ) h ( τ, g ( t )) dτ. (44)We provide two versions of the proof. The (cid:28)rst starts with the solution f ( x, t ) in Eq. (44) and veri(cid:28)es that itsatis(cid:28)es Eq. (43). The se ond starts from the partial integro-di(cid:27)erential equation (43) and derives the solution f ( x, t ) under ertain assumptions stated below Eq. (49).Proof 1: For (cid:28)rst we observe that L { f ( x, t ); g ( t ) , s } = 1 s e K ( s ) L {G ( x, t ); t, e K ( s ) − } . (45)With the hange of variables g ( s ) = z , we write: u ( x, g − ( w )) = u ( x ) + Z w K ( w − z ) P x u ( x, g − ( z )) dz, w = g ( t ) . (46)We want to show that Eq. (44) with the hoi e (42) solves Eq. (43). If we take the Lapla e transform of Eq.(43) using Eq. (46), we get: L { u ( x, t ); g ( t ) , s } = u ( x ) s + e K ( s ) P x L { u ( x, t ); g ( t ) , s } that is: s L { u ( x, t ); g ( t ) , s } − u ( x ) = s e K ( s ) P x L { u ( x, t ); g ( t ) , s } . (47)Now, if we substitute on Eq. (47) a solution of the form (44), u ( x, t ) = Z ∞ H ( x, τ ) h ( τ, g ( t )) dτ, (48)we have: e K ( s ) − L {H ( x, t ); t, e K ( s ) − } = u ( x ) + P x L {H ( x, t ); t, e K ( s ) − } τ e H ( x, τ ) = u ( x ) + P x e H ( x, τ ) , in whi h one readily re ognizes the Lapla e transform of the Markovian Fokker-Plan k equation with the sameinitial ondition u ( x ) . Therefore: ∂ t H ( x, t ) = P x H ( x, t ) , H ( x,
0) = u ( x ) . This argument shows not only that Eq. (44) is the fundamental solution of Eq. (43), but also that a generalsolution is given by Eq. (48) (see Corollary 4.1). This result is summarized in Corollary 6.1 (see below).Proof 2: We now start from Eq.(43) and we use integral transforms in order to get the fundamental solu-tion. Let F denote the Fourier transform operator and let: ( F ϕ )( k, t ) = b ϕ ( k, t ) = Z R e ikx ϕ ( x, t ) dx . Sin e b u ( k ) = 1 , and sin e ( FP x u )( k, t ) = ( FP x F − F u )( k, t ) = b P k b u ( k, t ) , where b P k = ( FP x F − ) k denotes theFourier transform of the operator P x , we have: b u ( k, g − ( w )) = 1 + Z w K ( w − z ) b P k b u ( k, g − ( z )) dz. Taking the Lapla e transform we have: L { b u ( k, g − ( w )); w, s } = s − + b P k e K ( s ) L { b u ( k, g − ( w )); w, s } , whi h is the same as: L { b u ( k, t ); g ( t ) , s } = s − + b P k e K ( s ) L { b u ( k, t ); g ( t ) , s } . Therefore: (cid:16) e K ( s ) − − b P k (cid:17) L { b u ( k, t ); g ( t ) , s } = s − e K ( s ) − . Denoting k ) = 1 , we have: L { b u ( k, t ); g ( t ) , s } = 1 s e K ( s ) (cid:16) e K ( s ) − − b P k (cid:17) − k ) , (49)where we suppose that the operator (cid:16) e K ( s ) − − b P k (cid:17) − is well de(cid:28)ned and a ts on the onstant fun tion k ) = 1 .Observe that the Fokker-Plan k equation (41) is obtained from Eq. (43) by setting K ( t ) = 1 , for ea h t ≥ ,that is e K ( s ) = s − , and g ( t ) = t , for ea h t ≥ . In this ase Eq. (49) be omes: L { b G ( k, t ); t, s } = ( s − b P k ) − k ) . (50)where G ( x, t ) is the fundamental solution. Taking the inverse Fourier transform, we get: L {G ( x, t ); t, s } = F − n ( s − b P k ) − k ) ; k, x o , (51)where: F − { ϕ ( k, s ) ; k, x } = 12 π Z R e − ikx ϕ ( k, s ) dk. (52)Repla ing s by e K ( s ) − in Eq. (51), one has: L {G ( x, t ); t, e K ( s ) − } = F − n ( e K ( s ) − − b P k ) − k ) ; k, x o . (53)12oing ba k to Eq. (49) and inverting the Fourier transform we obtain in view of Eq. (53): L { u ( x, t ); g ( t ) , s } = 1 s e K ( s ) F − (cid:26)(cid:16) e K ( s ) − − b P k (cid:17) − k ) ; k, x (cid:27) = 1 s e K ( s ) L {G ( x, t ); t, e K ( s ) − } . that is Eq. (45). (cid:3) Remark 6.1. If the Markovian pro ess is a Brownian motion one has P x = ∂ ∂x . The Fourier transform of P x is b P k = − k and Eq. (49) be omes: L { b u ( k, t ); g ( t ) , s } = 1 s e K ( s ) (cid:16) e K ( s ) − + k (cid:17) − k ) , where (cid:16) e K ( s ) − + k (cid:17) − k ) = 1 (cid:16) e K ( s ) − + k (cid:17) , whi h is well de(cid:28)ned be ause e K ( s ) − is positive.Corollary 6.1. If H ( x, t ) is a general solution of the Markovian Fokker-Plan k equation (41) with initial ondition H ( x,
0) = u ( x ) , then the fun tion: u ( x, t ) = Z ∞ H ( x, τ ) h ( τ, g ( t )) dτ (54)is a general solution of Eq. (43).>From a sto hasti point of view, f ( x, t ) ould be seen as the marginal distribution at time t of the subor-dinated pro ess: D ( t ) = Q ( l ( g ( t ))) (55)where Q is the di(cid:27)usion governed by the Fokker-Plan k equation (41) and l ( t ) is the random time pro ess,independent of Q ( t ) , with marginal distributions de(cid:28)ned by h ( τ, t ) .7 Examples involving standard Brownian motionIn the following examples, we onsider sto hasti models where the operator P x in Eq. (41) is ∂ xx , namelythe operator orresponding to standard Brownian motion. We will study more general operators in the nextse tion. We shall hoose various kernels K ( t ) and various stret hing fun tions g ( t ) . We let h ( τ, t ) denote thefundamental solution of the non-Markovian forward drift equation (24). Sin e the orresponding sto hasti models are not unique, we will mainly fo us on the subordinated pro ess B ( l ( t )) , t ≥ . However, we also giveexamples of other appropriate sto hasti models.7.1 Time-fra tional di(cid:27)usion equationLet g ( t ) = t . Consider the β -power kernel: K ( t ) = t β − Γ( β ) , < β ≤ , (56)and let h ( τ, t ) denote the fundamental solution of the non-Markovian forward drift equation (24) with kernel(56).Remark 7.1. In view of Theorem 3.1, su h a kernel arises if one requires h ( τ, t ) to be the marginal densityfun tion of a self-similar random time pro ess l ( t ) of order β .13nsertingEq. 56) in Eq. (37) we obtain the following equation: u ( x, t ) = u ( t ) + 1Γ( β ) Z t ( t − s ) β − ∂ xx u ( x, s ) ds , (57)whi h is sometimes alled the time-fra tional di(cid:27)usion equation [27,44℄. In view of Theorem 4.1, the fundamentalsolution is: f ( x, t ) = Z ∞ G ( x, τ ) h ( τ, t ) dτ, where h ( τ, t ) satis(cid:28)es: L { h ( τ, t ); t, s } = s β − e − τs β , τ, s ≥ . (58)Su h a fun tion h ( τ, t ) an be expressed as: h ( τ, t ) = t − β M β ( τ t − β ) , (59)where M β ( r ) , is de(cid:28)ned for < β < by the power series [24, 25℄: M β ( r ) = ∞ X k =0 ( − r ) k k !Γ [ − βk + (1 − β )]= 1 π ∞ X k =0 ( − r ) k k ! Γ [( β ( k + 1))] sin [ πβ ( k + 1)] , r ≥ . (60)The above series de(cid:28)nes a trans endental fun tion (entire of order / (1 − β ) ) [12℄. τ t=1 M β ( τ ) β =1/4 β =1/2 β =3/4 Figure 1: Plot of the density fun tion h ( τ, t ) = t − β M ( τ t − β ) at time t = 1 , for di(cid:27)erent values of the parameter β = [1 / , / , / . 14emark 7.2. The fun tion h ( τ, t ) in Eq. (59) represents the fundamental solution of the time-fra tionalforward drift equation (see also [15℄): u ( τ, t ) = u ( τ ) − β ) Z t ( t − s ) β − ∂ τ u ( τ, s ) ds. (61)This equation redu es to the standard drift equation when β → .7.1.1 Properties of the M -fun tionIt is useful to re all some important properties of the M -fun tion [12, 29℄. These are best expressed in terms ofthe fun tion M β ( τ, t ) = t − β M β ( τ t − β ) , (62)de(cid:28)ned for any τ, t ≥ and < β < .1. The Lapla e transform of M β ( τ, t ) with respe t to t is: L {M β ( τ, t ); t, s } = s β − e − τs β , τ, s ≥ . (63)2. The above equation suggests that in the singular limit β → one has: M ( τ, t ) = δ ( τ − t ) , τ, t ≥ . (64)3. If β = 1 / : M / ( τ, t ) = 1 √ πt exp( − τ / t ) , τ, t ≥ . (65)4. The M -fun tion is a parti ular ase of a Fox H -fun tion [30, 44℄. We indi ate with M { ϕ ( x ); x, u } = Z ∞ ϕ ( x ) x u − dx, (66)the Mellin transform of a fun tion ϕ ( x ) , x ≥ , with respe t to x evaluated in u ≥ . The Fox H -fun tion H m,np,q ( z ) = H m,np,q (cid:18) z (cid:12)(cid:12)(cid:12) ( a i , α i ) i =1 ,...,p ( b j , β j ) j =1 ,...,q (cid:19) , is hara terized by its Mellin transform as follows: M { H m,np,q ( z ); z, u } = A ( u ) B ( u ) C ( u ) D ( u ) , (67)with A ( u ) = m Y i =1 Γ( b j + β j u ) , B ( u ) = n Y j =1 Γ(1 − a j − α j u ) ,C ( u ) = q Y i = m +1 Γ(1 − b j − β j u ) , D ( u ) = p Y j = n +1 Γ( a j + α j u ) . Here: ≤ m ≤ q , ≤ n ≤ p , α j , β j > and a j , b j ∈ C (see [11, 33, 46℄ for more details).Starting from Eq. (63) and skipping to the Mellin transform, it is easy to show that we have the followingrelation: M β ( τ, t ) = t − β H , , (cid:18) τ t − β (cid:12)(cid:12)(cid:12) (1 − β, β )(0 , (cid:19) , τ, t ≥ , < β < . (68)15. Using the representation Eq. (68) and Eq. (67) we have for any η, β ∈ (0 , , see also [29℄: M ν ( x, t ) = Z ∞ M η ( x, τ ) M β ( τ, t ) dτ, ν = ηβ x ≥ . (69)The expression (59) for the fun tion h ( τ, t ) follows from Eq. (63), that is: h ( τ, t ) = M β ( τ, t ) , τ, t ≥ . (70)Moreover, when β → ,Eq. 64) gives h ( τ, t ) = δ ( τ − t ) as expe ted (see Remark 7.2). Comparing Eq. (9) andEq. (65) one observes that: G ( x, t ) = 12 M / ( | x | , t ) . (71)Using Theorem 4.1 and Eq. (69) together with Eq. (70) and Eq. (71) we re over the fundamental solution ofthe time-fra tional di(cid:27)usion equation [27℄: f ( x, t ) = Z ∞ G ( x, τ ) h ( τ, t ) d τ = 12 Z ∞ M / ( | x | , τ ) M β ( τ, t ) dτ = 12 M β/ ( | x | , t ) = 12 t − β/ M β/ ( | x | t − β/ ) . (72)Several plots of the M -fun tion are presented: in Figure 1 the fun tion h ( τ, t ) = M β ( τ, t ) is drawn at a (cid:28)xedtime t = 1 and for di(cid:27)erent values of the parameter β ; in Figure 2 is presented the plot of f ( x, t ) = M β/ ( | x | , t ) at a (cid:28)xed time t = 1 and for di(cid:27)erent values of β ; in Figure 3 is shown the time evolution of f ( x, t ) for (cid:28)xed β = 1 / . −5 −4 −3 −2 −1 0 1 2 3 4 500.10.20.30.40.5 x t=1 f(x,t) β =1/4 β =1/2 β =3/4 β =1 Figure 2: Plot of the density fun tion f ( x, t ) given by Eq. (72) at time t = 1 , for di(cid:27)erent values of theparameter β = [1 / , / , / , . For β = 1 one re overs the standard Gaussian density (71).16 x β =1/2 f(x,t) t=0.1t=1t=10t=10 Figure 3: Plot of the density fun tion f ( x, t ) for (cid:28)xed β = 1 / , at di(cid:27)erent times t = [0 . , , , ] .7.1.2 Sto hasti interpretations of the solutionFrom a sto hasti point of view, the fun tion h ( τ, t ) in Eq. (59) an be regarded as the marginal distribution of l β ( t ) , t ≥ , where l β ( t ) , t ≥ , is an H -ss random time with H = β . We have that for ea h integer m ≥ : E ( l β ( t ) m ) = m !Γ( βm + 1) t βm . (73)In fa t, from Eq. (58), for ea h integer m ≥ , we have : Z ∞ τ m s β − e − τs β dτ = m ! s − mβ − , whi h, inverting the Lapla e transform, gives Eq. (73).For instan e, with the suitable onventions [8℄, l β ( t ) , t ≥ , an be viewed as the lo al time in zero at time t of a d = 2(1 − β ) -dimensional Bessel pro ess [37℄. The fun tion f ( x, t ) in Eq. (72) is then the marginal densityfun tion of D ( t ) = B ( l β ( t )) , whi h is self-similar with H = β/ . In this ase, be ause l β ( t ) is self-similar of order β , we immediately havean example of a di(cid:27)erent pro ess with the same marginal distribution of D ( t ) (see Example 5.1). In fa t, ifwe onsider a (cid:16)standard(cid:17) fra tional Brownian motion B β/ of order β/ , then f ( x, t ) an also be seen as themarginal distribution of Y ( t ) = q l β (1) B β/ ( t ) , (74)17 −2 −1 log(t)B(l (t))l (t)log σ (t) β =1/2 Figure 4: Traje tory of the pro ess B ( l β ( t )) (top panel), with < t < and β = 1 / . The random time pro essis hosen to be l / ( t ) = | b ( t ) | where b ( t ) is a (cid:16)standard(cid:17) Brownian motion (see Example 1.1). The orrespondingtraje tory of the random time pro ess is presented in the middle panel. The estimated varian e, omputed ona sample of dimension N = 5000 , is presented in logarithmi s ale in the bottom panel and (cid:28)ts perfe tly thetheoreti al urve t / / Γ(3 / .where B β/ ( t ) is assumed to be independent of l β (1) (see Example 5.1). The pro ess Y ( t ) , t ≥ , is alled greyBrownian motion [43℄.From Eq. (73) one an derive immediately all the moments for the pro esses D ( t ) and Y ( t ) . For any integer m ≥ E ( D ( t ) m +1 ) = E ( Y ( t ) m +1 ) = 0; E ( D ( t ) m ) = E ( Y ( t ) m ) = 2 m !Γ( βm + 1) t βm . (75)Be ause < β < , the varian e grows slower than linearly with respe t to time. In this ase one speaks aboutslow anomalous di(cid:27)usion. Moreover, the in rements of the fra tional Brownian motion B β/ ( t ) do not havelong-range dependen e. In ontrast, the next example allows for the presen e of long-range dependen e throughthe introdu tion of a s aling fun tion g ( t ) = t α/β (see also Example 1.2).7.2 (cid:16)Stret hed(cid:17) time-fra tional di(cid:27)usion equationIf in the setup of Se tion 7.1, where the kernel K ( t ) is given by Eq. (56), we introdu e a s aling time g ( t ) = t α/β β =1/2t=1 Figure 5: Marginal density fun tion f ( x, t ) = M / ( | x | , t ) of the pro ess B ( l / ( t )) at time t = 1 and x ∈ [ − , . The histogram is evaluated over N = 10 simulated traje tories of the pro ess B ( | b ( t ) | ) (Figure 4).with α > , then the integral equation (57) is repla ed by Eq. (37), namely u ( x, t ) = u ( t ) + 1Γ( β ) αβ Z t s αβ − (cid:16) t αβ − s αβ (cid:17) β − ∂ xx u ( x, s ) ds. (76)Therefore, using Eq. 59): h ( τ, g ( t )) = g ( t ) − β M β ( τ g ( t ) − β ) = t − α M β ( τ t − α ) , and, using Eq. (72), the fundamental solution f ( x, t ) of Eq. (76) reads: f ( x, t ) = f ( x, g ( t )) = 12 t − α/ M β/ ( | x | t − α/ ) , t ≥ . (77)The fun tion f ( x, t ) , t ≥ , is the marginal distribution of the pro ess D ( t ) = B (cid:16) l β ( t α/β )) (cid:17) , t ≥ . The time- hange pro ess l β ( t α/β ) is self-similar of order H = α and the pro ess D ( t ) is then self-similar with H = α/ . In the ase < α < , the fun tion f ( x, t ) is also the marginal density of Y ( t ) = q l β (1) B α/ ( t ) , t ≥ , < α < , (78)where B α/ ( t ) is a (cid:16)standard(cid:17) fBm of order H = α/ independent of l β (1) . The pro ess Y ( t ) , t ≥ , is alledgeneralized grey Brownian motion [38℄. 19 −1 −2 log(t)−5 −4 −3 −2 −1 0 1 2 3 4 500.10.20.30.40.5 x (l (1)) B (t) σ (t)f(x,t) β =1/2 α =3/2t=1 Figure 6: Traje tory of the pro ess p l β (1) B α/ ( t ) (top panel), with < t < , β = 1 / and α = 3 / . Therandom variable l / (1) is Gaussian, see Eq. (65). The estimated varian e, omputed on a sample of dimension N = 5000 , is presented in logarithmi s ale in the middle panel together with the theoreti al urve t / / Γ(3 / .In the bottom panel the histogram, evaluated over a sample of N = 10 traje tories, (cid:28)ts the exa t marginaldensity Eq. (77) at time t = 1 .In this ase, for any integer m ≥ : E ( D ( t ) m +1 ) = E ( Y ( t ) m +1 ) = 0; E ( D ( t ) m ) = E ( Y ( t ) m ) = 2 m !Γ( βm + 1) t αm . (79)We have slow di(cid:27)usion when < α < (the varian e grows slower than linearly in time) and fast di(cid:27)usionwhen < α < (the varian e grows faster than linearly in time). In this ase the in rements of the pro ess Y ( t ) exhibit long-range dependen e.7.3 Exponential-de ay kernelLet g ( t ) = t . With the exponential-de ay kernel: K ( t ) = exp( − at ) , a ≥ , t ≥ , (80)we obtain the following equation: u ( x, t ) = u ( x ) + Z t e − a ( t − s ) ∂ xx u ( x, s ) ds. (81)In this ase e K ( s ) = ( s + a ) − and the marginal distribution of the random time pro ess l ( t ) , t ≥ , is de(cid:28)nedby Eq. (26): L { f l ( τ, t ); t, s } = s + as e − τ ( s + a ) , τ ≥ . τ l ( τ ,t) a=1 Figure 7: Plots of the marginal density of the random time f l ( τ, t ) Eq. 82) as a fun tion of τ at times t =[0 . , , . , and with a = 1 . The verti al line orresponds to a point mass (delta fun tion).Therefore, f l ( τ, t ) = e − τa ( δ ( τ − t ) + aθ ( t − τ )) = e − ta δ ( τ − t ) + ae − τa θ ( t − τ ) , (82)where θ ( x ) is the step fun tion (28). A graphi al representation of the time evolution of f l ( τ, t ) is presented inFigure 7.Remark 7.3. The fun tion f l ( τ, t ) de(cid:28)ned in Eq. 82) is the fundamental solution, in the sense of distributions,of the (cid:16)exponential(cid:17) forward drift equation: u ( τ, t ) = u ( τ ) − Z t e − a ( t − s ) ∂ τ u ( τ, s ) ds. This follows from Proposition 3.1. To he k it dire tly we note that f l ( τ,